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The fuzzy Sylvester matrix equation

System of simultaneous matrix equations plays a major role in various areas such as mathematics, physics, statistics, engineering, and social sciences. In many problems in various areas of science, which can be solved by solving a linear matrix equation, some of the system parameters are vague or imprecise, and fuzzy mathematics is better than crisp mathematics for mathematical modeling of these problems, and hence solving a linear matrix equation where some or all elements of the system are fuzzy is important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [

Since Friedman et al. [

The LR fuzzy number and its arithmetic operations were first introduced by Dubois and Prade. We know that triangular fuzzy numbers are just specious cases of LR fuzzy numbers. In particular, Allahviranloo et al. [

In Section

A fuzzy number is a fuzzy set like

Supp

Let

The definition of a right shape function

An LR fuzzy number

Clearly,

Also, two LR fuzzy numbers

Noticing that

(1) If

(2) If

(3) If

Based on the extension principle, the arithmetic operations for LR fuzzy numbers were defined. For arbitrary LR fuzzy numbers

addition

scalar multiplication

Let

For a subset

For a system of linear equations

Suppose

A matrix

For example, we represent

The matrix system

Using matrix notation, we have

An LR fuzzy numbers matrix

In this section we investigate the fuzzy Sylvester matrix equations (

At first, we convert the fuzzy Sylvester matrix equation (

Let

Let

Let

Setting

We combine (

The matrix

Applying the extension operation to two sides of (

For simplicity, we denote

Secondly, we extend the fuzzy LR linear system (

The LR fuzzy linear system (

Let

Let

Suppose

In order to solve the LR fuzzy Sylvester matrix equation (

The following Lemma shows when the matrix

The matrix

It seems that we have obtained the solution of the original fuzzy linear matrix system (

Let

When linear equation (

For instance, we seek

Now we define the LR fuzzy approximate solution of the fuzzy matrix equations (

Let

Let

It seems that the solution of the LR fuzzy linear system (

Now we give a sufficient condition for LR fuzzy solution to the fuzzy Sylvester matrix equation.

To illustrate the expression (

Let

The key points to make the solution vector be an LR fuzzy solution are

By the above analysis, one has the following conclusion.

Let

By further study, one gives a sufficient condition for obtaining nonnegative LR fuzzy solution of fuzzy Sylvester matrix equation (

Let

Since

Now that

The following theorems give some results for such

The inverse of a nonnegative matrix

The matrix

Let

There exists a permutation matrix

Consider the following fuzzy matrix system:

From Theorem

The coefficient matrices

Since

According to Theorems

Consider the fuzzy Sylvester matrix system

By Theorems

From Theorem

The coefficient matrices

Since

According to Theorems

In this work we presented a model for solving fuzzy Sylvester matrix equations

The authors were very thankful for the reviewer’s helpful suggestions to improve the paper. This paper was financially supported by the National Natural Science Foundation of China (71061013) and the Youth Scientific Research Ability Promotion Project of Northwest Normal University (nwnu-lkqn-11-20).